the Klein bottle is gotten from a quotient in the two-torus via
so to get all the isotopy classes one need to consider the auto-homeomorphisms of the torus.
It is well known that each of those are determined by a two-by-two matrix of , i.e where , so in seeking those matrices which obey the above gluing conditions we are compeled to analyse
for each couple .
Then if we set this implies and , so is odd. Also, if then and hence .
But , then are the only matrices which preserve the gluing conditions.
It is easy to see that these four matrices form the famous 4-Klein group. So .